If `vecaxx(vecbxxvecc)=vecbxx(veccxxveca)` and `[(vec, vecb, vecc)]!=0`
then `vecaxx(vecbxxvecc)` is equal to

Options

A. `vec0`
B. `vecaxxvecb`
C. `vecbxxvecc`
D. `veccxxveca`

Explanation

To solve the problem, we need to evaluate the expression \( \vec{a} \times (\vec{b} \times \vec{c}) \) under the condition that \( \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} \times (\vec{c} \times \vec{a}) \) and that \( [\vec{a}, \vec{b}, \vec{c}] \neq 0 \).

### Step-by-Step Solution:

1. **Understanding the Vector Triple Product**:
The vector triple product can be expressed using the identity:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
and similarly,
\[ \vec{b} \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a} \]

2. **Setting Up the Equation**:
Given the equality:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} \times (\vec{c} \times \vec{a}) \]
we can substitute the expressions from the vector triple product identities:
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a} \]

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} = (\vec{a} \cdot \vec{b}) \vec{c} - (\vec{b} \cdot \vec{a}) \vec{c} \]
This simplifies to:
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = 0 \]

4. **Analyzing the Result**:
For the above equation to hold true, we can conclude that:
\[ (\vec{a} \cdot \vec{c}) \vec{b} = (\vec{a} \cdot \vec{b}) \vec{c} \]
This implies that either \( \vec{b} \) and \( \vec{c} \) are parallel (which contradicts the condition \( [\vec{a}, \vec{b}, \vec{c}] \neq 0 \)), or \( \vec{a} \cdot \vec{c} = 0 \) and \( \vec{a} \cdot \vec{b} = 0 \).

5. **Final Conclusion**:
Therefore, we conclude that:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = 0 \]
Hence, the value of \( \vec{a} \times (\vec{b} \times \vec{c}) \) is equal to \( \vec{0} \).

### Final Answer:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = \vec{0} \]